Modeling of Granular Materials

Modeling of Granular MaterialsAbhinav GolasCOMP 768 - Physically Based SimulationApril 23, 20091MotivationMovies, games

Engineering design grain silosAvalanches, LandslidesApril 23, 20092

Spiderman 3The Mummywww.stheoutlawtorn.comOverviewWhat are Granular Materials?SimulationRenderingApril 23, 20093OverviewWhat are Granular Materials?SimulationRenderingApril 23, 20094What are Granular Materials?A granular material is a conglomeration of discrete solid, macroscopic particles characterized by a loss of energy whenever the particles interact (Wikipedia)Size variation from 1m to icebergsFood grains, sand, coal etc.Powders can be suspended in gasApril 23, 20095What are Granular materials?Can exist similar to various forms of matterGas/Liquid powders can be carried by velocity fieldsSandstormsLiquid/Solid similar to liquids embedded with multiple solid objectsAvalanches, landslidesHourglassSimilar to viscous liquidsApril 23, 20096Why the separate classification?Behavior not consistent with any one state of matterCan sustain small shear stresses stable pilesHydrostatic pressure achieves a maximumParticle interactions lose energyCollisions approach inelasticInfinite collisions in finite time inelastic collapseInhomogeneous and anisotropicParticle shape and size inhomogeneousApril 23, 20097Granular solids, liquids, and gases Jaeger et al.Understanding the behavior - StressStress

At equilibrium matrix is symmetric 6 degrees of freedomPressure for fluids tr()/ 3

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yzyzShearNormalStressDifferent matrix for different basis need invariantsPressure! I0Deviatoric invariants Invariants based on J1,J2Eigen values? called principle stressesApril 23, 20099

Understanding the behaviorWhy can sand sustain shear stress?Friction between particlesWhen does it yield? yield surface/conditionApril 23, 200910

Yield surfaceMany surfaces suitable for different materialsMohr Coulomb surface with Von-Mises equivalent stress f(I0, J1)

Condition for stability/rigidity:

sin coefficient of friction

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So why is it difficult to simulate?Scale - >10M particlesNonlinear behavior yield surfaceRepresentation discrete or continuum?April 23, 200912OverviewWhat are Granular Materials?SimulationRenderingApril 23, 200913SimulationDepends on what scenario to simulate2 dimensional Animating Sand, Mud, and Snow, Sumner et al.Discrete particles Particle-Based Simulation of Granular Materials, Bell et al.Continuum Animating Sand as a Fluid, Zhu et al.April 23, 200914Animating Sand, Mud, and SnowModel deformations on a 2D height field surface caused by rigid bodiesHash based grid space savingModel featuresMaterial redistribution, compressionParticles that get stuck to rigid bodyApril 23, 200915

Animating Sand, Mud, and SnowRigid body intersection check ray castingExtra materialDisplacedCompressedTransferring extra materialConstruct distance field to nearest clear cellTransfer material to that cellApril 23, 200916

Redistribution of materialErosionDistribute material equally to all neighboring lower height cells if slope > thresholdParticle GenerationMaterial may get stuck to bottom of bodySeed particle system from each contact triangle on rigid body volume = c * area of triangleVolume lost per time step exponential decayApril 23, 200917

Particle-Based Simulation of Granular MaterialsUse a particle system with collision handlingDefine objects in terms of spheresNeed to define per sphere pair interaction forcesCollision system based on Molecular DynamicsAllow minor spatial overlap between objects

April 23, 200918Sphere pair interactionDefine overlap(), relative velocity(V), contact normal(N), normal and tangential velocities(Vn, Vt), rate of change of overlap(V.N)Normal forces

kd : dissipation during collisions, kr : particle stiffnessBest choice of coefficients: =1/2, =3/2Given coefficient of restitution , and time of contact tc, we can determine kd and kr

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Sphere pair interactionTangential forces

These forces cannot stop motion require true static frictionSprings between particles with persistent contact?Non-spherical objectsApril 23, 200920

Solid bodiesMap mesh to structure built from spheresGenerate distance field from meshChoose offset from mesh to place spheresBuild iso-surface mesh (Marching Tetrahedra)Sample spheres randomly on trianglesLet them float to desired iso-surface by repulsion forcesD=sphere density, A=triangle area, R=particle radius, place particles, 1 more with fractional probabilityApril 23, 200921

Solid bodiesK interaction kernel, P Position of particle, V velocity of particle, distance fieldRigid body evolutionOverall force = forcesOverall torque = torques around center of massApril 23, 200922

Efficient collision detectionSpatial hashingGrid size = 2 x Maximum particle radiusNeed to look at 27 cells for each particle O(n)Not good enough, insert each particle into not 1, but 27 cells check only one cell for possible collisionsWhy better?Spatial coherenceParticles moving to next grid cell, rare (inelastic collapse)Wonderful for stagnant regionsApril 23, 200923Advantages/DisadvantagesThe GoodFaithful to actual physical behaviorThe Bad and the UglyComputationally intensiveSmall scale scenesScenes with some control particlesApril 23, 200924Animating Sand as a FluidMotivationSand ~ viscous fluids in some casesContinuum simulationBootstrap additions to existing fluid simulatorWhy?Simulation independent of number of particlesBetter numerical stability than rigid body simulatorsApril 23, 200925Fluid simulation? whats that?Discretize 3D region into cuboidal grid3 step process to solve Navier Stokes equationsAdvectAdd body forcesIncompressibility projectionStable and accurate under CFL conditionApril 23, 200926vvuup, Extending our fluid simulatorExtra things we need for sandFriction (internal, boundary)Rigid portions in sandRecallStressYield conditionApril 23, 200927Calculating stressExact calculation infeasibleSmart approximationsDefine strain rate D = d/dt(strain)

Approximate stressesRigidFluidApril 23, 200928

The algorithm in a nutshellCalculate strain rateFind rigid stress for cellCell satisfies yield condition?Yes mark rigid, store rigid stressNo mark fluid, store fluid stressFor each rigid connected componentAccumulate forces and torquesFor fluid cells, subtract friction forceApril 23, 200929Yield conditionRecap

Can add a cohesive force for sticky materialsApril 23, 200930

Rigid componentsAll velocities must lie in allowed space of rigid motion (D=0)Find connected components graph searchAccumulate momentum and angular momentum

Ri solid region, u velocity, density, I moment of inertiaApril 23, 200931

Rigid Fluid: Animating the Interplay Between Rigid Bodies and Fluid, Carlson et al.Friction in fluid cellsUpdate cell velocityBoundary conditionsNormal velocity: Tangential velocity:

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RepresentationDefining regions of sandLevel setsParticlesAllow improved advectionHybrid simulationPIC Particle In CellFLIP FLuid Implicit ParticleApril 23, 200933AdvectionSemi Lagrangian advectionDissipativeRelies on incompressibility volume conservationHybrid approachUse grids coupled with particlesAdvect particles no averaging losses!April 23, 200934PIC and FLIP methodsPIC Particle in cellParticles support gridParticles take velocity from gridFLIP Fluid Implicit ParticleGrid supports particlesParticles take acceleration from gridApril 23, 200935PIC and FLIP methodsNo grid based advectionLesser dissipationParticle advection is simplerNo need for a level setPIC, more dissipative suited for viscous flows, FLIP for inviscid flowsSurface reconstruction?April 23, 200936Surface reconstructionSurface (x) define using all particles i

k(s) kernel function, R 2 x average particle spacing, x position, r radius, wi weight for particleSuitable choice of kernel?Must provide flat surfaces for what should be flatk(s) =max(0,(1-s2)3)April 23, 200937

Surface reconstructionIssuesConcave regions centre might lie outside regionSmoothing passRadii must be close approximation to distance from surfaceNon-trivial, constant particle radius assumedAdvantagesFastNo temporal interdependenceApril 23, 200938Advantages/DisadvantagesAdvantagesFast & stableIndependent of number of particles large scale scenes possibleDisadvantagesNot completely true to actual behaviorDetail issues smoothing in simulation, surface reconstructionApril 23, 200939OverviewWhat are Granular Materials?SimulationRenderingApril 23, 200940RenderingNon-trivial due to scale and visual complexitySurface based renderingUse volumetric texturesTexture advected by fluid velocityParticle renderingApril 23, 200941Particle RenderingLevel of detail necessary

Different approaches for different levelsApril 23, 200942

Rendering Tons of Sand, Sony Pictures ImageworksParticle RenderingSand cloudsLight Reflection Functions for Simulation of Clouds and Dusty Surfaces, BlinnDefines lighting and scattering functions for such materialsSuitable options for dust, cloudsApril 23, 200943Rendering Tons of SandSurfaces with sandGenerate particles on mesh at runtime with temporal coherenceSand particlesGenerate required number for visual detail from base control particlesRendering level of detail

Peak of 480 million particles at render timeApril 23, 200944ConclusionsInteresting, albeit difficult problemModels not perfectSpeed vs. scale/realism tradeoffSimilar tradeoff in renderingApril 23, 200945ReferencesGranular Solids, Liquids, and Gases Jaeger et al.Review of Modern Physics 96Particle-Based Simulations of Granular Materials Bell et al., Eurographics 05Animating Sand as a Fluid Zhu et al.SIGGRAPH 05Rigid Fluid: Animating the Interplay Between Rigid Bodies and Fluid Carlson et al.SIGGRAPH 04April 23, 200946ReferencesRendering Tons of Sand Allen et al. Sony Pictures ImageworksSIGGRAPH 2007 sketchesA Fast Variational Framework for Accurate Solid-Fluid Coupling Batty et al.SIGGRAPH 2007Animating Sand, Mud, and Snow Sumner et al.Eurographics 1999April 23, 200947